On a combination of the 1-2-3 Conjecture and the Antimagic Labelling Conjecture

Abstract : This paper is dedicated to studying the following question: Is it always possible to injectively assign the weights 1, ..., |E(G)| to the edges of any given graph G (with no component isomorphic to K2) so that every two adjacent vertices of G get distinguished by their sums of incident weights? One may see this question as a combination of the well-known 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Throughout this paper, we exhibit evidence that this question might be true. Benefiting from the investigations on the Antimagic Labelling Conjecture, we first point out that several classes of graphs, such as regular graphs, indeed admit such assignments. We then show that trees also do, answering a recent conjecture of Arumugam, Premalatha, Bača and Semaničová-Feňovčíková. Towards a general answer to the question above, we then prove that claimed assignments can be constructed for any graph, provided we are allowed to use some number of additional edge weights. For some classes of sparse graphs, namely 2-degenerate graphs and graphs with maximum average degree 3, we show that only a small (constant) number of such additional weights suffices.
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Article dans une revue
Discrete Mathematics and Theoretical Computer Science, DMTCS, 2017, Vol 19 no. 1
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https://hal.archives-ouvertes.fr/hal-01361482
Contributeur : Julien Bensmail <>
Soumis le : lundi 7 août 2017 - 14:29:20
Dernière modification le : mercredi 9 août 2017 - 01:08:19

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Julien Bensmail, Mohammed Senhaji, Kasper Szabo Lyngsie. On a combination of the 1-2-3 Conjecture and the Antimagic Labelling Conjecture. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2017, Vol 19 no. 1. <hal-01361482v3>

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